Airfoil shape for turbine rotor blades

ABSTRACT

A turbine rotor blade having an airfoil that includes a pressure side portion of a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of a pressure side as set forth in Table I. The Cartesian coordinate values of X, Y, and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil defined along the Z axis. The X and Y values of the pressure side are coordinate values that, when connected by smooth continuing arcs, define pressure side sections of the pressure side portion of the nominal airfoil profile at each Z coordinate value. The pressure side sections may be joined smoothly with one another to form the pressure side portion.

BACKGROUND OF THE INVENTION

The subject matter disclosed herein relates to turbine engine airfoils and, more specifically, to airfoils of turbine rotor blades.

Some aircraft and/or power plant systems, for example certain jet aircraft, gas turbines, and combined cycle power plant systems, employ turbines (also referred to as turbomachines) in their design and operation. Some of these turbines employ airfoils (e.g., turbine rotor blades, blades, airfoils, etc.) which during operation are exposed to fluid flows. These airfoils are configured to aerodynamically interact with the fluid flows and generate energy (e.g., creating thrust, turning kinetic energy to mechanical energy, thermal energy to mechanical energy, etc.) from these fluid flows as part of power generation. As a result of this interaction and conversion, the aerodynamic characteristics and losses of these airfoils have an impact on system and turbine operation, performance, thrust, efficiency, and power.

BRIEF DESCRIPTION OF THE INVENTION

Aspects and advantages are set forth below in the following description, or may be obvious from the description, or may be learned through practice.

According to an exemplary embodiment, the present disclosure includes a turbine rotor blade having an airfoil that includes a pressure side portion of a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of a pressure side as set forth in Table I. The Cartesian coordinate values of X, Y, and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil defined along the Z axis. The X and Y values of the pressure side are coordinate values that, when connected by smooth continuing arcs, define pressure side sections of the pressure side portion of the nominal airfoil profile at each Z coordinate value. The pressure side sections may be joined smoothly with one another to form the pressure side portion.

According to another exemplary embodiment, the present disclosure includes a turbine rotor blade having an airfoil that includes a suction side portion of a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of a suction side as set forth in Table I. The Cartesian coordinate values of X, Y, and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil defined along the Z axis. The X and Y values of the suction side are coordinate values that, when connected by smooth continuing arcs, define suction side sections of the suction side portion of the nominal airfoil profile at each Z coordinate value. The suction side sections may be joined smoothly with one another to form the suction side portion.

According to another exemplary embodiment, the present disclosure includes a turbine engine that has an airfoil that includes: a pressure side portion of a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of a pressure side as set forth in Table I; and a suction side portion of the nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of the suction side as set forth in Table I. The Cartesian coordinate values of X, Y, and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil defined along the Z axis. The X and Y values of the pressure side are coordinate values that, when connected by smooth continuing arcs, define pressure side sections of the pressure side portion of the nominal airfoil profile at each Z coordinate value. The pressure side sections may be joined smoothly with one another to form the pressure side portion. The X and Y values of the suction side are coordinate values that, when connected by smooth continuing arcs, define suction side sections of the suction side portion of the nominal airfoil profile at each Z coordinate value. The suction side sections may be joined smoothly with one another to form the suction side portion.

Those of ordinary skill in the art will better appreciate the features and aspects of such embodiments, and others, upon review of the specification.

BRIEF DESCRIPTION OF THE DRAWINGS

A full and enabling disclosure of various embodiments, including the best mode thereof to one skilled in the art, is set forth more particularly in the remainder of the specification, including reference to the accompanying figures, in which:

FIG. 1 is a schematic representation of an exemplary turbine having multiple stages with each stage including alternating rows of turbine rotor blades and stationary vanes or nozzles according to at least one embodiment of the present disclosure; and

FIG. 2 is a perspective view of a turbine rotor blade according to at least one embodiment of the present disclosure.

DETAILED DESCRIPTION OF THE INVENTION

One or more specific embodiments of the present subject matter will be described below. In an effort to provide a concise description of these embodiments, all features of an actual implementation may not be described in the specification. It should be appreciated that in the development of any such actual implementation, as in any engineering or design project, numerous implementation-specific decisions must be made to achieve the developers' specific goals, such as compliance with system-related and business-related constraints, which may vary from one implementation to another. Moreover, it should be appreciated that such a development effort might be complex and time consuming, but would nevertheless be a routine undertaking of design, fabrication, and manufacture for those of ordinary skill having the benefit of this disclosure.

When introducing elements of various embodiments of the present subject matter, the articles “a,” “an,” “the,” and “said” are intended to mean that there are one or more of the elements. The terms “comprising,” “including,” and “having” are intended to be inclusive and mean that there may be additional elements other than the listed elements.

Referring now to the drawings, particularly to FIG. 1, there is illustrated an exemplary turbine 10 of a combustion or gas turbine 12 including a plurality of turbine stages arranged in serial flow order. Each stage of the turbine includes a row of turbine vanes or nozzles disposed axially adjacent to a corresponding row of turbine rotor blades. While four stages are illustrated in FIG. 1, the exact number of stages of the turbine 10 is a choice of engineering design, and may be more or less than the four stages shown. The four stages are merely exemplary of one turbine design, and are not intended to limit the presently claimed turbine rotor blade in any manner.

As depicted in FIG. 1, the first stage includes a plurality of turbine nozzles 100 and a plurality of turbine rotor blades 150. The turbine nozzles 100 are annularly arranged about an axis of a turbine rotor 14, while the turbine rotor blades 150 are annularly arranged about and coupled to the turbine rotor 14. A second stage of the turbine 12 includes a plurality of turbine nozzles 200 annularly arranged about the axis of the turbine rotor 14 and a plurality of turbine rotor blades 250 annularly arranged about and coupled to the turbine rotor 14. A third stage of the turbine 12 includes a plurality of turbine nozzles 300 annularly arranged about the axis of the turbine rotor 14 and a plurality of turbine rotor blades 350 annularly arranged about and coupled to the turbine rotor 14. In particular embodiments, the turbine 12 may further include a fourth stage, which also includes plurality of turbine nozzles 400 annularly arranged about the axis of the turbine rotor 14 and a plurality of turbine rotor blades 450 annularly arranged about and coupled to the turbine rotor 14. As will be appreciated, the turbine nozzles 100, 200, 300, 400 may be coupled to the casing 16 of the turbine 12 and remain stationary during operation.

It will be appreciated that the turbine nozzles 100, 200, 300 and 400 and turbine rotor blades 150, 250, 350 and 450 are disposed or at least partially disposed within a working fluid or hot gas path 18 of the turbine 12. The various stages of the turbine 10 at least partially define the hot gas path 18 through which combustion gases, as indicated by arrows 20, flow during operation of the gas turbine 12.

FIG. 2 provides a perspective view of an exemplary turbine rotor blade 50, as may be incorporated in one of the stages of the turbine 12. In a preferred embodiment of the present disclosure, the turbine rotor blade 50 (as described by the points included within Table I below) is a turbine rotor blade 150 of the plurality of the turbine rotor blades 150 used in the first stage of the turbine 12, as shown in FIG. 1. As illustrated in FIG. 2, the turbine rotor blade 50 includes an airfoil 52 having a pressure side 54 and an opposing suction side 56. The pressure side 54 and the suction side 56 meet or intersect at a leading edge 58 and a trailing edge 60 of the airfoil 52. A chord line 62 extends between the leading edge 58 and the trailing edge 60 such that pressure and suction sides 54, 56 can be said to extend in chord or chordwise between the leading edge 58 and the trailing edge 60.

The airfoil 52 further includes a first end or root 64 which intersects with and extends radially outwardly from a base or platform 66 of the turbine rotor blade 50. The airfoil 52 terminates radially at a second end or radial tip 68 of the airfoil 52. The pressure and suction sides 54, 56 can be said to extend in span or in a span-wise direction 70 (along the height of the airfoil 52) between the root 64 and/or the platform 66 and the radial tip 68 of the airfoil 52. In other words, each turbine rotor blade 50 includes an airfoil 52 having opposing pressure and suction sides 54, 56 that extend in chord or chordwise 62 between opposing leading and trailing edges 58, 60 and that extend in span or span-wise 70 between the root 64 and the radial tip 68 of the airfoil 52.

In particular configurations, the airfoil 52 may include a fillet 72 formed between the platform 66 and the airfoil 52 proximate to the root 64. The fillet 72 can include a weld or braze fillet, which can be formed via conventional MIG welding, TIG welding, brazing, etc., and can include a profile that reduces fluid dynamic losses. In particular embodiments, the platform 66, airfoil 52 and the fillet 72 can be formed as a single component, such as by casting and/or machining and/or 3D printing and/or any other suitable technique now known or later developed and/or discovered. In particular configurations, the turbine rotor blade 50 includes a mounting portion 74 which is formed to connect and/or to secure the turbine rotor blade 50 to the rotor shaft 14.

The airfoil 52 of the turbine rotor blade 50 has a profile at any cross-section taken between the platform 66 or the root 64 and the radial tip 68. In accordance with the present disclosure, the X, Y, and Z values of the profile are given in Table I as percentage values of the airfoil span or height. As one example only, the height of the airfoil 52 of the rotor blade 50 may be from about 2 inches to about 50 inches. As another example, the height of the airfoil 52 of the rotor blade 50 may be from about 3 inches to about 10 inches. However, it is to be understood that heights below or above this range may also be employed as desired in the specific application.

A hot gas path of a gas turbine requires airfoils that meet system requirements of aerodynamic and mechanical blade loading and efficiency. To define the airfoil shape of each turbine rotor blade airfoil, there is a unique set or loci of points in space that meet the stage requirements and that can be manufactured. This unique loci of points meets the requirements for stage efficiency and are arrived at by iteration between aerodynamic and mechanical loadings enabling the turbine to run in an efficient, safe and smooth manner. These points are unique and specific to the system.

The loci that defines the airfoil profile of turbine rotor blade 50 includes a set of points with X, Y and Z dimensions relative to a reference origin coordinate system, as provided in Table I and shown in FIG. 2, respectively. The Cartesian coordinate system of X, Y and Z values given in Table I below defines the airfoil profile of the turbine rotor blade 50 at various locations along its length or span or, as used herein, height. Table I lists data for a non-coated airfoil, and, as discussed more below, the envelope/tolerance for the coordinates may be about +/−5% in a direction normal to any airfoil surface location and/or about +/−5% of the chord 62 in a direction nominal to any airfoil surface location. As shown in FIG. 2, the point data origin 76 is defined at or proximate to the root 64 at the leading edge 58 of the airfoil 52. In one embodiment, as presented in Table I below, the point data origin 76 is defined at or proximate to (above or below) a transition or intersection line 78 defined between the fillet 72 and the airfoil 52. The point data origin 76 corresponds to the non-dimensional Z value of Table I (presented below) at Z equals 0.

The coordinate values for the X, Y and Z coordinates are set forth in non-dimensionalized units in Table I, although units of dimensions may be used when the values are appropriately converted. That is, the X, Y, and Z values set forth in Table I are expressed in non-dimensional form (X, Y, and Z) from 0% to 100% of the radial span 70 or height of the airfoil. As one example, the Cartesian coordinate values of X, Y and Z may be convertible to dimensional distances by multiplying the X, Y and Z values by a height of the airfoil at the leading edge 58 and/or multiplying by a constant number. As another example, the Cartesian coordinate values of X, Y and Z may be convertible to dimensional distances by multiplying the X, Y and Z values by a height of the airfoil at the trailing edge 60 and/or multiplying by a constant number. Thus, to convert the Z value to a Z coordinate value, for example, in inches, the non-dimensional Z value given in Table I is multiplied by the height of the airfoil in inches.

As described above, the Cartesian coordinate system has orthogonally-related X, Y and Z axes, where the X axis lies generally parallel to a centerline of the rotor shaft 14, i.e., the rotary axis and a positive X coordinate value is axial toward an aft, i.e., exhaust end, of the turbine 10. The positive Y coordinate value extends tangentially in the direction of rotation of the rotor, and the positive Z coordinate value extends radially outwardly toward the radial tip 68 of the airfoil 52. All the values in Table I are given at room temperature and do not include the fillet 72.

By defining X and Y coordinate values at selected locations in a Z direction normal to the X, Y plane, the airfoil shape or profile sections (which may be referred to as “pressure side sections” on the pressure side and “suction side sections” on the suction side) of the airfoil 52 of the turbine rotor blade 50 along the span or height of the airfoil 52 can be ascertained. Thus, by connecting the X and Y values with smooth continuing arcs, each profile section (which may include a pressure side section and suction side section) at each distance Z is determined. The airfoil profiles of the various surface locations between the distances Z can then be determined by smoothly connecting the adjacent profile sections to one another to form the airfoil profile.

The Table I values are generated and shown to four decimal places for determining the profile of the airfoil. As the turbine rotor blade heats up during operation of the gas turbine, mechanical stresses and elevated temperatures will cause a change in the X, Y and Z values. Accordingly, it should be understood that the values for the nominal airfoil profile given in Table I represent ambient, non-operating or non-hot conditions (e.g., room temperature) and are for an uncoated airfoil.

There are typical manufacturing tolerances as well as coatings which may be accounted for in the actual profile of the airfoil 52. It will therefore be appreciated that +/− typical manufacturing tolerances, i.e., +/− values, including any coating thicknesses, may be additive to the X and Y values given in Table I below. Accordingly, a distance of +/− 5% in a direction normal to any airfoil surface location and/or about +/− 5% of the chord 62 in a direction nominal to any airfoil surface location may define an airfoil profile envelope for this particular airfoil design, i.e., a range of variation between measured points on the actual airfoil surface at nominal cold or room temperature and the ideal position of those points as given in Table I below at the same temperature. According to another example, a tolerance of about 10-20 percent of a thickness of the airfoil's trailing edge 60 in a direction normal to any airfoil surface location may define a range of variation between measured points on an actual airfoil surface and ideal positions as embodied by the invention in Table I. As should further be understood, the data provided in Table I is scalable and the geometry pertains to all aerodynamic scales and/or RPM ranges. The design of the airfoil 52 for the turbine rotor blade 50 is robust to this range of variation without impairment of mechanical and aerodynamic functions.

TABLE I SUCTION SIDE PRESSURE SIDE X (%) Y (%) Z (%) X (%) Y (%) Z (%) 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 −0.5321 0.5897 0.0000 0.4882 −0.3373 0.0000 −0.8804 1.3055 0.0000 1.0285 −0.5842 0.0000 −0.9572 1.6963 0.0000 1.3165 −0.6624 0.0000 −1.0340 2.0872 0.0000 1.6045 −0.7405 0.0000 −0.9380 3.3159 0.0000 2.5205 −0.8009 0.0000 −0.6418 4.5199 0.0000 3.4311 −0.6829 0.0000 −0.2633 5.6992 0.0000 4.3307 −0.4663 0.0000 0.1947 6.8484 0.0000 5.2110 −0.1947 0.0000 0.7186 7.9702 0.0000 6.0805 0.1097 0.0000 1.3028 9.0590 0.0000 6.9417 0.4388 0.0000 1.9418 10.1204 0.0000 7.7946 0.7789 0.0000 2.6275 11.1516 0.0000 8.6476 1.1300 0.0000 3.3488 12.1555 0.0000 9.4978 1.4838 0.0000 4.1030 13.1291 0.0000 10.3508 1.8321 0.0000 4.8847 14.0781 0.0000 11.2038 2.1749 0.0000 5.6938 14.9996 0.0000 12.0649 2.5068 0.0000 6.5303 15.8910 0.0000 12.9261 2.8277 0.0000 7.3942 16.7549 0.0000 13.7956 3.1321 0.0000 8.2883 17.5887 0.0000 14.6705 3.4228 0.0000 9.2098 18.3950 0.0000 15.5481 3.6971 0.0000 10.1615 19.1739 0.0000 16.4340 3.9522 0.0000 11.1434 19.9199 0.0000 17.3254 4.1880 0.0000 12.1527 20.6357 0.0000 18.2195 4.4047 0.0000 13.1867 21.3104 0.0000 19.1191 4.5994 0.0000 14.2399 21.9495 0.0000 20.0241 4.7722 0.0000 15.3150 22.5419 0.0000 20.9320 4.9231 0.0000 16.4093 23.0904 0.0000 21.8453 5.0465 0.0000 17.5283 23.5951 0.0000 22.7586 5.1452 0.0000 18.6747 24.0476 0.0000 23.6774 5.2165 0.0000 19.8486 24.4480 0.0000 24.5989 5.2632 0.0000 21.0389 24.7881 0.0000 25.5177 5.2796 0.0000 22.2457 25.0624 0.0000 26.4392 5.2659 0.0000 23.4552 25.2708 0.0000 27.3580 5.2275 0.0000 24.6784 25.4107 0.0000 28.2768 5.1589 0.0000 25.9126 25.4711 0.0000 29.1928 5.0602 0.0000 27.1550 25.4573 0.0000 30.1061 4.9368 0.0000 28.3892 25.3668 0.0000 31.0140 4.7832 0.0000 29.6125 25.1995 0.0000 31.9163 4.6049 0.0000 30.8192 24.9609 0.0000 32.8131 4.3965 0.0000 32.0095 24.6510 0.0000 33.7045 4.1634 0.0000 33.1861 24.2698 0.0000 34.5876 3.9028 0.0000 34.3435 23.8172 0.0000 35.4625 3.6148 0.0000 35.4735 23.3071 0.0000 36.3292 3.3022 0.0000 36.5678 22.7449 0.0000 37.1849 2.9676 0.0000 37.6320 22.1305 0.0000 38.0324 2.6055 0.0000 38.6687 21.4723 0.0000 38.8689 2.2216 0.0000 39.6808 20.7674 0.0000 39.6945 1.8129 0.0000 40.6681 20.0214 0.0000 40.5090 1.3823 0.0000 41.6280 19.2397 0.0000 41.3126 0.9325 0.0000 42.5633 18.4279 0.0000 42.1053 0.4635 0.0000 43.4684 17.5887 0.0000 42.8842 −0.0274 0.0000 44.3515 16.7275 0.0000 43.6521 −0.5348 0.0000 45.2100 15.8471 0.0000 44.4064 −1.0614 0.0000 46.0465 14.9420 0.0000 45.1496 −1.6072 0.0000 46.8638 14.0177 0.0000 45.8819 −2.1667 0.0000 47.6619 13.0770 0.0000 46.6005 −2.7427 0.0000 48.4408 12.1171 0.0000 47.3053 −3.3323 0.0000 49.2033 11.1434 0.0000 47.9992 −3.9385 0.0000 49.9493 10.1561 0.0000 48.6794 −4.5583 0.0000 50.6788 9.1605 0.0000 49.3486 −5.1918 0.0000 51.3919 8.1539 0.0000 50.0069 −5.8364 0.0000 52.0913 7.1419 0.0000 50.6514 −6.4946 0.0000 52.7797 6.1189 0.0000 51.2849 −7.1638 0.0000 53.4516 5.0876 0.0000 51.9075 −7.8440 0.0000 54.1126 4.0509 0.0000 52.5164 −8.5324 0.0000 54.7626 3.0060 0.0000 53.1143 −9.2345 0.0000 55.4044 1.9528 0.0000 53.7012 −9.9449 0.0000 56.0325 0.8968 0.0000 54.2772 −10.6607 0.0000 56.6496 −0.1700 0.0000 54.8422 −11.3875 0.0000 57.2584 −1.2397 0.0000 55.3962 −12.1225 0.0000 57.8536 −2.3175 0.0000 55.9420 −12.8658 0.0000 58.4405 −3.4036 0.0000 56.4795 −13.6145 0.0000 59.0165 −4.4925 0.0000 57.0061 −14.3688 0.0000 59.5815 −5.5895 0.0000 57.5245 −15.1285 0.0000 60.1410 −6.6893 0.0000 58.0374 −15.8964 0.0000 60.6895 −7.7946 0.0000 58.5393 −16.6671 0.0000 61.2325 −8.9054 0.0000 59.0357 −17.4433 0.0000 61.7674 −10.0189 0.0000 59.5239 −18.2250 0.0000 62.2912 −11.1352 0.0000 60.0038 −19.0094 0.0000 62.8068 −12.2569 0.0000 60.4756 −19.7992 0.0000 63.3115 −13.3814 0.0000 60.9418 −20.5946 0.0000 63.8052 −14.5114 0.0000 61.3999 −21.3927 0.0000 64.2851 −15.6441 0.0000 61.8496 −22.1963 0.0000 64.7569 −16.7823 0.0000 62.2885 −23.0027 0.0000 65.2204 −17.9260 0.0000 62.7218 −23.8172 0.0000 65.6757 −19.0724 0.0000 63.1442 −24.6373 0.0000 66.1254 −20.2189 0.0000 63.5556 −25.4601 0.0000 66.5698 −21.3708 0.0000 63.9588 −26.2884 0.0000 67.0058 −22.5254 0.0000 64.3537 −27.1194 0.0000 67.4337 −23.6828 0.0000 64.7432 −27.9532 0.0000 67.8561 −24.8430 0.0000 65.1216 −28.7924 0.0000 68.2702 −26.0031 0.0000 65.4919 −29.6344 0.0000 68.6871 −27.1633 0.0000 65.8567 −30.4819 0.0000 69.1122 −28.3234 0.0000 66.2242 −31.3294 0.0000 69.5510 −29.4781 0.0000 66.6219 −32.1631 0.0000 69.9871 −30.6355 0.0000 67.1018 −32.9558 0.0000 70.3245 −31.8176 0.0000 67.7244 −33.6359 0.0000 70.2833 −33.0298 0.0000 68.5417 −34.0062 0.0000 69.4358 −33.9047 0.0000 69.4358 −33.9047 0.0000 1.3988 1.1903 9.9997 1.3988 1.1903 9.9997 0.8365 1.7471 9.9997 1.8952 0.8804 9.9997 0.4388 2.4300 9.9997 2.4355 0.6528 9.9997 0.3319 2.8099 9.9997 2.7193 0.5814 9.9997 0.2249 3.1897 9.9997 3.0032 0.5101 9.9997 0.2276 4.4157 9.9997 3.9083 0.4635 9.9997 0.4498 5.6252 9.9997 4.8024 0.5979 9.9997 0.7679 6.8128 9.9997 5.6773 0.8475 9.9997 1.1684 7.9784 9.9997 6.5303 1.1656 9.9997 1.6401 9.1111 9.9997 7.3668 1.5167 9.9997 2.1694 10.2137 9.9997 8.1951 1.8869 9.9997 2.7564 11.2805 9.9997 9.0206 2.2654 9.9997 3.3899 12.3200 9.9997 9.8407 2.6521 9.9997 4.0728 13.3348 9.9997 10.6634 3.0361 9.9997 4.7942 14.3222 9.9997 11.4862 3.4174 9.9997 5.5511 15.2848 9.9997 12.3173 3.7849 9.9997 6.3410 16.2146 9.9997 13.1538 4.1359 9.9997 7.1611 17.1114 9.9997 13.9985 4.4678 9.9997 8.0113 17.9781 9.9997 14.8515 4.7750 9.9997 8.8889 18.8091 9.9997 15.7127 5.0602 9.9997 9.7968 19.6127 9.9997 16.5821 5.3235 9.9997 10.7348 20.3807 9.9997 17.4570 5.5621 9.9997 11.7084 21.1157 9.9997 18.3401 5.7733 9.9997 12.7122 21.8151 9.9997 19.2288 5.9543 9.9997 13.7435 22.4733 9.9997 20.1229 6.1106 9.9997 14.8021 23.0849 9.9997 21.0197 6.2368 9.9997 15.8800 23.6472 9.9997 21.9221 6.3355 9.9997 16.9825 24.1573 9.9997 22.8271 6.4041 9.9997 18.1070 24.6153 9.9997 23.7349 6.4398 9.9997 19.2589 25.0158 9.9997 24.6428 6.4480 9.9997 20.4383 25.3586 9.9997 25.5478 6.4206 9.9997 21.6341 25.6329 9.9997 26.4557 6.3657 9.9997 22.8381 25.8386 9.9997 27.3580 6.2779 9.9997 24.0449 25.9702 9.9997 28.2576 6.1600 9.9997 25.2599 26.0251 9.9997 29.1544 6.0146 9.9997 26.4858 26.0031 9.9997 30.0431 5.8419 9.9997 27.7091 25.8989 9.9997 30.9289 5.6389 9.9997 28.9213 25.7179 9.9997 31.8066 5.4140 9.9997 30.1144 25.4601 9.9997 32.6788 5.1617 9.9997 31.2855 25.1337 9.9997 33.5427 4.8847 9.9997 32.4429 24.7388 9.9997 34.3984 4.5830 9.9997 33.5783 24.2753 9.9997 35.2459 4.2593 9.9997 34.6891 23.7514 9.9997 36.0851 3.9138 9.9997 35.7642 23.1754 9.9997 36.9162 3.5463 9.9997 36.8065 22.5529 9.9997 37.7362 3.1595 9.9997 37.8157 21.8919 9.9997 38.5480 2.7509 9.9997 38.8031 21.1870 9.9997 39.3462 2.3230 9.9997 39.7630 20.4492 9.9997 40.1360 1.8760 9.9997 40.7010 19.6758 9.9997 40.9177 1.4125 9.9997 41.6143 18.8722 9.9997 41.6856 0.9298 9.9997 42.5084 18.0385 9.9997 42.4426 0.4333 9.9997 43.3806 17.1827 9.9997 43.1914 −0.0823 9.9997 44.2308 16.3051 9.9997 43.9291 −0.6116 9.9997 45.0618 15.4110 9.9997 44.6532 −1.1574 9.9997 45.8737 14.5004 9.9997 45.3690 −1.7169 9.9997 46.6636 13.5734 9.9997 46.0739 −2.2874 9.9997 47.4397 12.6327 9.9997 46.7650 −2.8743 9.9997 48.1994 11.6782 9.9997 47.4480 −3.4722 9.9997 48.9427 10.7073 9.9997 48.1199 −4.0838 9.9997 49.6750 9.7282 9.9997 48.7781 −4.7064 9.9997 50.3908 8.7408 9.9997 49.4282 −5.3400 9.9997 51.0957 7.7453 9.9997 50.0672 −5.9845 9.9997 51.7868 6.7414 9.9997 50.6953 −6.6400 9.9997 52.4670 5.7322 9.9997 51.3151 −7.3037 9.9997 53.1335 4.7146 9.9997 51.9212 −7.9784 9.9997 53.7917 3.6944 9.9997 52.5191 −8.6586 9.9997 54.4390 2.6659 9.9997 53.1088 −9.3497 9.9997 55.0753 1.6319 9.9997 53.6875 −10.0518 9.9997 55.7006 0.5897 9.9997 54.2552 −10.7594 9.9997 56.3177 −0.4553 9.9997 54.8120 −11.4753 9.9997 56.9238 −1.5057 9.9997 55.3578 −12.1993 9.9997 57.5217 −2.5644 9.9997 55.8953 −12.9289 9.9997 58.1087 −3.6258 9.9997 56.4247 −13.6667 9.9997 58.6846 −4.6954 9.9997 56.9430 −14.4099 9.9997 59.2524 −5.7678 9.9997 57.4559 −15.1614 9.9997 59.8119 −6.8457 9.9997 57.9606 −15.9129 9.9997 60.3604 −7.9263 9.9997 58.4597 −16.6726 9.9997 60.9034 −9.0151 9.9997 58.9507 −17.4351 9.9997 61.4382 −10.1067 9.9997 59.4361 −18.2030 9.9997 61.9648 −11.2038 9.9997 59.9133 −18.9737 9.9997 62.4832 −12.3063 9.9997 60.3823 −19.7499 9.9997 62.9933 −13.4143 9.9997 60.8431 −20.5315 9.9997 63.4925 −14.5279 9.9997 61.2929 −21.3214 9.9997 63.9779 −15.6441 9.9997 61.7345 −22.1140 9.9997 64.4552 −16.7659 9.9997 62.1651 −22.9122 9.9997 64.9214 −17.8931 9.9997 62.5847 −23.7158 9.9997 65.3767 −19.0258 9.9997 62.9933 −24.5276 9.9997 65.8265 −20.1585 9.9997 63.3938 −25.3421 9.9997 66.2708 −21.2940 9.9997 63.7832 −26.1595 9.9997 66.7124 −22.4349 9.9997 64.1644 −26.9823 9.9997 67.1485 −23.5731 9.9997 64.5402 −27.8078 9.9997 67.5791 −24.7168 9.9997 64.9132 −28.6361 9.9997 68.0042 −25.8578 9.9997 65.2835 −29.4644 9.9997 68.4238 −27.0015 9.9997 65.6537 −30.2926 9.9997 68.8407 −28.1451 9.9997 66.0377 −31.1154 9.9997 69.2713 −29.2861 9.9997 66.4573 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5.6060 19.9995 8.4803 19.0286 19.9995 15.5344 5.9324 19.9995 9.3442 19.8651 19.9995 16.3819 6.2313 19.9995 10.2411 20.6604 19.9995 17.2376 6.5028 19.9995 11.1681 21.4201 19.9995 18.1043 6.7442 19.9995 12.1280 22.1387 19.9995 18.9764 6.9526 19.9995 13.1264 22.8162 19.9995 19.8568 7.1254 19.9995 14.1576 23.4525 19.9995 20.7455 7.2653 19.9995 15.2190 24.0366 19.9995 21.6368 7.3723 19.9995 16.3051 24.5660 19.9995 22.5309 7.4436 19.9995 17.4131 25.0405 19.9995 23.4278 7.4820 19.9995 18.5486 25.4546 19.9995 24.3274 7.4847 19.9995 19.7060 25.8029 19.9995 25.2242 7.4518 19.9995 20.8881 26.0854 19.9995 26.1183 7.3860 19.9995 22.0784 26.2966 19.9995 27.0124 7.2872 19.9995 23.2714 26.4337 19.9995 27.9010 7.1556 19.9995 24.4700 26.4941 19.9995 28.7842 6.9910 19.9995 25.6768 26.4776 19.9995 29.6591 6.7990 19.9995 26.8863 26.3816 19.9995 30.5313 6.5796 19.9995 28.0875 26.2061 19.9995 31.3952 6.3328 19.9995 29.2669 25.9565 19.9995 32.2509 6.0613 19.9995 30.4243 25.6356 19.9995 33.0984 5.7651 19.9995 31.5652 25.2462 19.9995 33.9376 5.4469 19.9995 32.6870 24.7936 19.9995 34.7687 5.1041 19.9995 33.7868 24.2807 19.9995 35.5887 4.7421 19.9995 34.8509 23.7130 19.9995 36.4005 4.3581 19.9995 35.8849 23.1041 19.9995 37.2041 3.9577 19.9995 36.8832 22.4514 19.9995 37.9968 3.5353 19.9995 37.8596 21.7602 19.9995 38.7812 3.0965 19.9995 38.8058 21.0334 19.9995 39.5546 2.6412 19.9995 39.7329 20.2765 19.9995 40.3171 2.1694 19.9995 40.6352 19.4893 19.9995 41.0713 1.6812 19.9995 41.5183 18.6720 19.9995 41.8145 1.1766 19.9995 42.3795 17.8328 19.9995 42.5496 0.6610 19.9995 43.2243 16.9688 19.9995 43.2736 0.1289 19.9995 44.0498 16.0912 19.9995 43.9867 −0.4169 19.9995 44.8561 15.1971 19.9995 44.6888 −0.9764 19.9995 45.6433 14.2865 19.9995 45.3827 −1.5469 19.9995 46.4140 13.3650 19.9995 46.0657 −2.1283 19.9995 47.1682 12.4270 19.9995 46.7404 −2.7207 19.9995 47.9115 11.4780 19.9995 47.4041 −3.3268 19.9995 48.6383 10.5181 19.9995 48.0596 −3.9412 19.9995 49.3541 9.5499 19.9995 48.7041 −4.5665 19.9995 50.0590 8.5708 19.9995 49.3376 −5.2001 19.9995 50.7501 7.5862 19.9995 49.9657 −5.8446 19.9995 51.4330 6.5933 19.9995 50.5801 −6.4974 19.9995 52.1022 5.5950 19.9995 51.1862 −7.1611 19.9995 52.7632 4.5912 19.9995 51.7841 −7.8303 19.9995 53.4132 3.5847 19.9995 52.3738 −8.5105 19.9995 54.0550 2.5699 19.9995 52.9497 −9.1961 19.9995 54.6858 1.5496 19.9995 53.5202 −9.8900 19.9995 55.3084 0.5266 19.9995 54.0797 −10.5921 19.9995 55.9228 −0.5046 19.9995 54.6310 −11.2997 19.9995 56.5262 −1.5414 19.9995 55.1740 −12.0156 19.9995 57.1213 −2.5808 19.9995 55.7088 −12.7396 19.9995 57.7055 −3.6258 19.9995 56.2354 −13.4664 19.9995 58.2815 −4.6762 19.9995 56.7510 −14.2015 19.9995 58.8492 −5.7322 19.9995 57.2612 −14.9393 19.9995 59.4059 −6.7936 19.9995 57.7631 −15.6825 19.9995 59.9572 −7.8577 19.9995 58.2595 −16.4340 19.9995 60.5003 −8.9246 19.9995 58.7477 −17.1882 19.9995 61.0378 −9.9997 19.9995 59.2277 −17.9452 19.9995 61.5672 −11.0776 19.9995 59.7021 −18.7077 19.9995 62.0855 −12.1609 19.9995 60.1711 −19.4729 19.9995 62.5957 −13.2498 19.9995 60.6319 −20.2435 19.9995 63.0948 −14.3441 19.9995 61.0844 −21.0197 19.9995 63.5857 −15.4412 19.9995 61.5260 −21.8014 19.9995 64.0630 −16.5464 19.9995 61.9621 −22.5858 19.9995 64.5292 −17.6572 19.9995 62.3845 −23.3784 19.9995 64.9900 −18.7680 19.9995 62.7959 −24.1765 19.9995 65.4453 −19.8843 19.9995 63.1963 −24.9801 19.9995 65.8951 −21.0033 19.9995 63.5857 −25.7919 19.9995 66.3421 −22.1250 19.9995 63.9642 −26.6038 19.9995 66.7837 −23.2468 19.9995 64.3345 −27.4211 19.9995 67.2198 −24.3685 19.9995 64.6993 −28.2411 19.9995 67.6476 −25.4930 19.9995 65.0586 −29.0639 19.9995 68.0727 −26.6202 19.9995 65.4206 −29.8867 19.9995 68.4951 −27.7447 19.9995 65.7991 −30.7040 19.9995 68.9312 −28.8692 19.9995 66.2187 −31.5022 19.9995 69.3920 −29.9882 19.9995 66.7233 −32.2482 19.9995 69.7979 −31.1237 19.9995 67.3624 −32.8845 19.9995 69.8088 −32.3058 19.9995 68.1715 −33.2629 19.9995 69.0354 −33.1916 19.9995 69.0354 −33.1916 19.9995 3.1458 3.5408 29.9992 3.1458 3.5408 29.9992 2.6028 4.0893 29.9992 3.6203 3.2226 29.9992 2.2078 4.7530 29.9992 4.1359 2.9730 29.9992 2.0926 5.1205 29.9992 4.4088 2.8798 29.9992 1.9775 5.4881 29.9992 4.6817 2.7865 29.9992 1.9281 6.6811 29.9992 5.5566 2.6275 29.9992 2.0844 7.8659 29.9992 6.4370 2.6412 29.9992 2.3532 9.0288 29.9992 7.3010 2.8249 29.9992 2.7043 10.1753 29.9992 8.1375 3.1239 29.9992 3.1129 11.2943 29.9992 8.9520 3.4859 29.9992 3.5764 12.3831 29.9992 9.7501 3.8754 29.9992 4.0920 13.4445 29.9992 10.5345 4.2895 29.9992 4.6598 14.4785 29.9992 11.3135 4.7146 29.9992 5.2714 15.4850 29.9992 12.0896 5.1452 29.9992 5.9296 16.4669 29.9992 12.8685 5.5676 29.9992 6.6317 17.4268 29.9992 13.6584 5.9762 29.9992 7.3723 18.3593 29.9992 14.4565 6.3630 29.9992 8.1539 19.2534 29.9992 15.2656 6.7223 29.9992 8.9685 20.1119 29.9992 16.0884 7.0541 29.9992 9.8215 20.9320 29.9992 16.9249 7.3531 29.9992 10.7101 21.7136 29.9992 17.7724 7.6164 29.9992 11.6371 22.4541 29.9992 18.6281 7.8467 29.9992 12.6052 23.1508 29.9992 19.4948 8.0415 29.9992 13.6036 23.7980 29.9992 20.3670 8.2005 29.9992 14.6293 24.3932 29.9992 21.2446 8.3185 29.9992 15.6825 24.9307 29.9992 22.1278 8.4008 29.9992 16.7659 25.4135 29.9992 23.0136 8.4446 29.9992 17.8821 25.8386 29.9992 23.9023 8.4529 29.9992 19.0176 26.1951 29.9992 24.7881 8.4227 29.9992 20.1668 26.4831 29.9992 25.6740 8.3541 29.9992 21.3269 26.6970 29.9992 26.5544 8.2527 29.9992 22.4980 26.8369 29.9992 27.4321 8.1183 29.9992 23.6856 26.9027 29.9992 28.3015 7.9510 29.9992 24.8786 26.8945 29.9992 29.1682 7.7507 29.9992 26.0635 26.8095 29.9992 30.0239 7.5231 29.9992 27.2318 26.6477 29.9992 30.8741 7.2708 29.9992 28.3892 26.4145 29.9992 31.7161 6.9883 29.9992 29.5329 26.1073 29.9992 32.5498 6.6839 29.9992 30.6629 25.7343 29.9992 33.3726 6.3547 29.9992 31.7709 25.2983 29.9992 34.1900 6.0037 29.9992 32.8515 24.8073 29.9992 34.9936 5.6334 29.9992 33.8992 24.2643 29.9992 35.7889 5.2412 29.9992 34.9195 23.6746 29.9992 36.5761 4.8298 29.9992 35.9151 23.0356 29.9992 37.3522 4.4047 29.9992 36.8887 22.3581 29.9992 38.1229 3.9604 29.9992 37.8377 21.6423 29.9992 38.8799 3.4996 29.9992 38.7620 20.8963 29.9992 39.6314 3.0252 29.9992 39.6616 20.1256 29.9992 40.3692 2.5342 29.9992 40.5365 19.3275 29.9992 41.1015 2.0296 29.9992 41.3867 18.5102 29.9992 41.8200 1.5112 29.9992 42.2205 17.6737 29.9992 42.5304 0.9819 29.9992 43.0350 16.8180 29.9992 43.2325 0.4388 29.9992 43.8304 15.9431 29.9992 43.9236 −0.1179 29.9992 44.6148 15.0544 29.9992 44.6066 −0.6857 29.9992 45.3800 14.1521 29.9992 45.2785 −1.2644 29.9992 46.1342 13.2333 29.9992 45.9422 −1.8513 29.9992 46.8693 12.3063 29.9992 46.5977 −2.4492 29.9992 47.5933 11.3683 29.9992 47.2450 −3.0553 29.9992 48.3037 10.4221 29.9992 47.8840 −3.6724 29.9992 49.0003 9.4649 29.9992 48.5121 −4.2977 29.9992 49.6887 8.5022 29.9992 49.1347 −4.9313 29.9992 50.3634 7.5341 29.9992 49.7490 −5.5703 29.9992 51.0299 6.5577 29.9992 50.3524 −6.2203 29.9992 51.6881 5.5758 29.9992 50.9476 −6.8786 29.9992 52.3381 4.5885 29.9992 51.5345 −7.5450 29.9992 52.9799 3.5956 29.9992 52.1105 −8.2197 29.9992 53.6134 2.5973 29.9992 52.6782 −8.8999 29.9992 54.2388 1.5962 29.9992 53.2377 −9.5883 29.9992 54.8559 0.5842 29.9992 53.7862 −10.2850 29.9992 55.4620 −0.4306 29.9992 54.3293 −10.9871 29.9992 56.0599 −1.4509 29.9992 54.8641 −11.6947 29.9992 56.6523 −2.4794 29.9992 55.3907 −12.4078 29.9992 57.2338 −3.5106 29.9992 55.9118 −13.1264 29.9992 57.8070 −4.5473 29.9992 56.4247 −13.8504 29.9992 58.3747 −5.5868 29.9992 56.9348 −14.5772 29.9992 58.9342 −6.6290 29.9992 57.4340 −15.3095 29.9992 59.4855 −7.6767 29.9992 57.9304 −16.0445 29.9992 60.0313 −8.7271 29.9992 58.4186 −16.7851 29.9992 60.5661 −9.7831 29.9992 58.8985 −17.5311 29.9992 61.0899 −10.8417 29.9992 59.3730 −18.2825 29.9992 61.6083 −11.9059 29.9992 59.8365 −19.0368 29.9992 62.1129 −12.9755 29.9992 60.2918 −19.7992 29.9992 62.6121 −14.0479 29.9992 60.7389 −20.5644 29.9992 63.1003 −15.1258 29.9992 61.1777 −21.3379 29.9992 63.5830 −16.2064 29.9992 61.6056 −22.1168 29.9992 64.0575 −17.2924 29.9992 62.0252 −22.8984 29.9992 64.5237 −18.3813 29.9992 62.4366 −23.6828 29.9992 64.9818 −19.4729 29.9992 62.8425 −24.4727 29.9992 65.4343 −20.5672 29.9992 63.2402 −25.2654 29.9992 65.8786 −21.6670 29.9992 63.6296 −26.0580 29.9992 66.3174 −22.7668 29.9992 64.0163 −26.8561 29.9992 66.7508 −23.8721 29.9992 64.3921 −27.6569 29.9992 67.1841 −24.9746 29.9992 64.7623 −28.4633 29.9992 67.6229 −26.0744 29.9992 65.1271 −29.2779 29.9992 68.0672 −27.1715 29.9992 65.4946 −30.0897 29.9992 68.5170 −28.2658 29.9992 65.8923 −30.8851 29.9992 68.9614 −29.3656 29.9992 66.3778 −31.6256 29.9992 69.3645 −30.4846 29.9992 67.0196 −32.2427 29.9992 69.4358 −31.6420 29.9992 67.8314 −32.6074 29.9992 68.6843 −32.5416 29.9992 68.6843 −32.5416 29.9992 3.7876 5.0904 39.9989 3.7876 5.0904 39.9989 3.2638 5.6389 39.9989 4.2456 4.7613 39.9989 2.8743 6.2862 39.9989 4.7475 4.4980 39.9989 2.7605 6.6468 39.9989 5.0122 4.3965 39.9989 2.6467 7.0075 39.9989 5.2769 4.2950 39.9989 2.5808 8.1759 39.9989 6.1298 4.0975 39.9989 2.7262 9.3415 39.9989 6.9993 4.0701 39.9989 2.9840 10.4824 39.9989 7.8605 4.2100 39.9989 3.3241 11.6042 39.9989 8.6942 4.4788 39.9989 3.7273 12.7067 39.9989 9.5006 4.8216 39.9989 4.1853 13.7873 39.9989 10.2904 5.2001 39.9989 4.6954 14.8405 39.9989 11.0694 5.6005 39.9989 5.2577 15.8608 39.9989 11.8400 6.0119 39.9989 5.8665 16.8481 39.9989 12.6107 6.4260 39.9989 6.5166 17.8081 39.9989 13.3842 6.8347 39.9989 7.2104 18.7378 39.9989 14.1658 7.2269 39.9989 7.9482 19.6402 39.9989 14.9612 7.5944 39.9989 8.7326 20.5096 39.9989 15.7648 7.9372 39.9989 9.5554 21.3379 39.9989 16.5848 8.2444 39.9989 10.4139 22.1250 39.9989 17.4159 8.5214 39.9989 11.3052 22.8683 39.9989 18.2551 8.7628 39.9989 12.2377 23.5649 39.9989 19.1053 8.9657 39.9989 13.2086 24.2149 39.9989 19.9665 9.1303 39.9989 14.2124 24.8156 39.9989 20.8305 9.2565 39.9989 15.2464 25.3559 39.9989 21.7026 9.3415 39.9989 16.3051 25.8358 39.9989 22.5775 9.3854 39.9989 17.3912 26.2555 39.9989 23.4525 9.3909 39.9989 18.5020 26.6093 39.9989 24.3246 9.3607 39.9989 19.6374 26.8945 39.9989 25.1968 9.2894 39.9989 20.7839 27.1057 39.9989 26.0662 9.1852 39.9989 21.9358 27.2401 39.9989 26.9301 9.0453 39.9989 23.0849 27.2977 39.9989 27.7886 8.8752 39.9989 24.2423 27.2812 39.9989 28.6388 8.6695 39.9989 25.4025 27.1907 39.9989 29.4808 8.4364 39.9989 26.5599 27.0289 39.9989 30.3173 8.1759 39.9989 27.7036 26.7930 39.9989 31.1429 7.8879 39.9989 28.8253 26.4941 39.9989 31.9602 7.5752 39.9989 29.9224 26.1293 39.9989 32.7693 7.2406 39.9989 31.0002 25.7042 39.9989 33.5674 6.8841 39.9989 32.0617 25.2242 39.9989 34.3573 6.5083 39.9989 33.1011 24.6894 39.9989 35.1362 6.1134 39.9989 34.1132 24.1080 39.9989 35.9069 5.6992 39.9989 35.0950 23.4854 39.9989 36.6693 5.2714 39.9989 36.0467 22.8271 39.9989 37.4235 4.8243 39.9989 36.9765 22.1332 39.9989 38.1668 4.3636 39.9989 37.8843 21.4092 39.9989 38.9018 3.8891 39.9989 38.7729 20.6549 39.9989 39.6286 3.3982 39.9989 39.6396 19.8760 39.9989 40.3445 2.8935 39.9989 40.4871 19.0752 39.9989 41.0493 2.3779 39.9989 41.3126 18.2579 39.9989 41.7460 1.8486 39.9989 42.1162 17.4214 39.9989 42.4344 1.3082 39.9989 42.9034 16.5711 39.9989 43.1118 0.7570 39.9989 43.6741 15.7045 39.9989 43.7838 0.1947 39.9989 44.4283 14.8268 39.9989 44.4447 −0.3757 39.9989 45.1661 13.9354 39.9989 45.0975 −0.9572 39.9989 45.8901 13.0331 39.9989 45.7448 −1.5496 39.9989 46.6032 12.1198 39.9989 46.3811 −2.1475 39.9989 47.3053 11.1983 39.9989 47.0119 −2.7564 39.9989 47.9965 10.2658 39.9989 47.6345 −3.3707 39.9989 48.6767 9.3250 39.9989 48.2488 −3.9961 39.9989 49.3486 8.3761 39.9989 48.8549 −4.6269 39.9989 50.0096 7.4216 39.9989 49.4501 −5.2659 39.9989 50.6624 6.4617 39.9989 50.0398 −5.9132 39.9989 51.3041 5.4935 39.9989 50.6212 −6.5659 39.9989 51.9404 4.5226 39.9989 51.1917 −7.2296 39.9989 52.5658 3.5463 39.9989 51.7567 −7.8961 39.9989 53.1856 2.5644 39.9989 52.3134 −8.5735 39.9989 53.7972 1.5798 39.9989 52.8620 −9.2537 39.9989 54.4006 0.5897 39.9989 53.4023 −9.9394 39.9989 54.9957 −0.4032 39.9989 53.9371 −10.6333 39.9989 55.5854 −1.4015 39.9989 54.4637 −11.3299 39.9989 56.1641 −2.4026 39.9989 54.9875 −12.0320 39.9989 56.7373 −3.4119 39.9989 55.5031 −12.7396 39.9989 57.3023 −4.4212 39.9989 56.0133 −13.4500 39.9989 57.8591 −5.4359 39.9989 56.5179 −14.1658 39.9989 58.4076 −6.4562 39.9989 57.0171 −14.8844 39.9989 58.9507 −7.4792 39.9989 57.5080 −15.6085 39.9989 59.4882 −8.5077 39.9989 57.9935 −16.3380 39.9989 60.0203 −9.5390 39.9989 58.4707 −17.0703 39.9989 60.5469 −10.5729 39.9989 58.9397 −17.8081 39.9989 61.0625 −11.6124 39.9989 59.4032 −18.5513 39.9989 61.5726 −12.6546 39.9989 59.8557 −19.2973 39.9989 62.0691 −13.7051 39.9989 60.3028 −20.0488 39.9989 62.5573 −14.7582 39.9989 60.7444 −20.8058 39.9989 63.0345 −15.8169 39.9989 61.1777 −21.5655 39.9989 63.5035 −16.8811 39.9989 61.6056 −22.3280 39.9989 63.9642 −17.9479 39.9989 62.0252 −23.0959 39.9989 64.4223 −19.0148 39.9989 62.4393 −23.8666 39.9989 64.8775 −20.0845 39.9989 62.8452 −24.6400 39.9989 65.3301 −21.1569 39.9989 63.2402 −25.4217 39.9989 65.7799 −22.2265 39.9989 63.6214 −26.2088 39.9989 66.2269 −23.2989 39.9989 63.9889 −27.0042 39.9989 66.6630 −24.3767 39.9989 64.3427 −27.8051 39.9989 67.0936 −25.4546 39.9989 64.6938 −28.6059 39.9989 67.5269 −26.5297 39.9989 65.0586 −29.4040 39.9989 67.9795 −27.6021 39.9989 65.4645 −30.1829 39.9989 68.4485 −28.6690 39.9989 65.9554 −30.9152 39.9989 68.8626 −29.7523 39.9989 66.5835 −31.5296 39.9989 68.9284 −30.8878 39.9989 67.3761 −31.8697 39.9989 68.2126 −31.7682 39.9989 68.2126 −31.7682 39.9989 4.2785 6.8786 49.9986 4.2785 6.8786 49.9986 3.7712 7.4216 49.9986 4.7311 6.5440 49.9986 3.3927 8.0634 49.9986 5.2275 6.2725 49.9986 3.2816 8.4186 49.9986 5.4881 6.1682 49.9986 3.1705 8.7738 49.9986 5.7486 6.0640 49.9986 3.1157 9.9174 49.9986 6.5906 5.8473 49.9986 3.2555 11.0556 49.9986 7.4518 5.7925 49.9986 3.5106 12.1774 49.9986 8.3102 5.9159 49.9986 3.8479 13.2717 49.9986 9.1385 6.1600 49.9986 4.2511 14.3414 49.9986 9.9421 6.4699 49.9986 4.7119 15.3863 49.9986 10.7293 6.8210 49.9986 5.2248 16.4093 49.9986 11.5082 7.1885 49.9986 5.7925 17.4049 49.9986 12.2816 7.5697 49.9986 6.4041 18.3676 49.9986 13.0550 7.9482 49.9986 7.0623 19.2946 49.9986 13.8285 8.3212 49.9986 7.7645 20.1860 49.9986 14.6129 8.6723 49.9986 8.5105 21.0444 49.9986 15.4028 8.9932 49.9986 9.3031 21.8672 49.9986 16.2036 9.2839 49.9986 10.1369 22.6516 49.9986 17.0100 9.5444 49.9986 11.0118 23.3894 49.9986 17.8273 9.7693 49.9986 11.9196 24.0778 49.9986 18.6555 9.9641 49.9986 12.8603 24.7168 49.9986 19.4948 10.1177 49.9986 13.8394 25.3037 49.9986 20.3478 10.2356 49.9986 14.8515 25.8358 49.9986 21.2090 10.3151 49.9986 15.8964 26.3076 49.9986 22.0729 10.3508 49.9986 16.9606 26.7162 49.9986 22.9368 10.3480 49.9986 18.0439 27.0536 49.9986 23.7953 10.3069 49.9986 19.1492 27.3251 49.9986 24.6455 10.2301 49.9986 20.2765 27.5280 49.9986 25.4930 10.1177 49.9986 21.4147 27.6542 49.9986 26.3350 9.9723 49.9986 22.5529 27.7008 49.9986 27.1742 9.7940 49.9986 23.6883 27.6734 49.9986 28.0108 9.5828 49.9986 24.8210 27.5719 49.9986 28.8418 9.3415 49.9986 25.9483 27.3991 49.9986 29.6618 9.0727 49.9986 27.0700 27.1578 49.9986 30.4737 8.7792 49.9986 28.1671 26.8534 49.9986 31.2718 8.4638 49.9986 29.2367 26.4886 49.9986 32.0589 8.1237 49.9986 30.2817 26.0690 49.9986 32.8351 7.7672 49.9986 31.3074 25.5972 49.9986 33.6030 7.3915 49.9986 32.3140 25.0761 49.9986 34.3627 6.9965 49.9986 33.2986 24.5084 49.9986 35.1115 6.5851 49.9986 34.2613 23.8995 49.9986 35.8547 6.1545 49.9986 35.2020 23.2522 49.9986 36.5870 5.7102 49.9986 36.1180 22.5721 49.9986 37.3138 5.2494 49.9986 37.0067 21.8672 49.9986 38.0324 4.7750 49.9986 37.8761 21.1349 49.9986 38.7428 4.2868 49.9986 38.7263 20.3779 49.9986 39.4449 3.7876 49.9986 39.5573 19.5990 49.9986 40.1360 3.2747 49.9986 40.3692 18.7982 49.9986 40.8190 2.7509 49.9986 41.1645 17.9809 49.9986 41.4937 2.2188 49.9986 41.9407 17.1498 49.9986 42.1601 1.6730 49.9986 42.6977 16.3051 49.9986 42.8184 1.1190 49.9986 43.4382 15.4466 49.9986 43.4656 0.5540 49.9986 44.1623 14.5772 49.9986 44.1074 −0.0192 49.9986 44.8753 13.6996 49.9986 44.7410 −0.6006 49.9986 45.5747 12.8109 49.9986 45.3663 −1.1931 49.9986 46.2631 11.9141 49.9986 45.9861 −1.7910 49.9986 46.9406 11.0090 49.9986 46.5977 −2.3998 49.9986 47.6098 10.0985 49.9986 47.2011 −3.0142 49.9986 48.2708 9.1797 49.9986 47.7990 −3.6368 49.9986 48.9208 8.2554 49.9986 48.3887 −4.2648 49.9986 49.5598 7.3229 49.9986 48.9701 −4.9011 49.9986 50.1906 6.3822 49.9986 49.5461 −5.5457 49.9986 50.8159 5.4387 49.9986 50.1138 −6.1929 49.9986 51.4303 4.4870 49.9986 50.6733 −6.8484 49.9986 52.0419 3.5298 49.9986 51.2273 −7.5067 49.9986 52.6453 2.5699 49.9986 51.7759 −8.1731 49.9986 53.2404 1.6045 49.9986 52.3162 −8.8451 49.9986 53.8329 0.6363 49.9986 52.8510 −9.5225 49.9986 54.4143 −0.3373 49.9986 53.3776 −10.2027 49.9986 54.9903 −1.3165 49.9986 53.8987 −10.8911 49.9986 55.5580 −2.2983 49.9986 54.4143 −11.5822 49.9986 56.1202 −3.2830 49.9986 54.9217 −12.2816 49.9986 56.6770 −4.2731 49.9986 55.4236 −12.9810 49.9986 57.2255 −5.2659 49.9986 55.9173 −13.6859 49.9986 57.7686 −6.2615 49.9986 56.4082 −14.3962 49.9986 58.3089 −7.2598 49.9986 56.8937 −15.1066 49.9986 58.8437 −8.2609 49.9986 57.3736 −15.8224 49.9986 59.3703 −9.2647 49.9986 57.8509 −16.5410 49.9986 59.8886 −10.2740 49.9986 58.3226 −17.2623 49.9986 60.4015 −11.2888 49.9986 58.7888 −17.9863 49.9986 60.9007 −12.3063 49.9986 59.2496 −18.7159 49.9986 61.3916 −13.3293 49.9986 59.7049 −19.4454 49.9986 61.8743 −14.3578 49.9986 60.1547 −20.1805 49.9986 62.3488 −15.3863 49.9986 60.5963 −20.9210 49.9986 62.8178 −16.4203 49.9986 61.0296 −21.6670 49.9986 63.2813 −17.4570 49.9986 61.4547 −22.4157 49.9986 63.7421 −18.4937 49.9986 61.8716 −23.1727 49.9986 64.1974 −19.5332 49.9986 62.2720 −23.9352 49.9986 64.6499 −20.5727 49.9986 62.6615 −24.7031 49.9986 65.0942 −21.6176 49.9986 63.0372 −25.4820 49.9986 65.5330 −22.6626 49.9986 63.3992 −26.2664 49.9986 65.9691 −23.7103 49.9986 63.7476 −27.0563 49.9986 66.4079 −24.7552 49.9986 64.0877 −27.8462 49.9986 66.8577 −25.7974 49.9986 64.4387 −28.6306 49.9986 67.3212 −26.8342 49.9986 64.8474 −29.3821 49.9986 67.7820 −27.8736 49.9986 65.3548 −30.0732 49.9986 68.1797 −28.9323 49.9986 65.9554 −30.6958 49.9986 68.2675 −30.0376 49.9986 66.7343 −31.0743 49.9986 67.5763 −30.9097 49.9986 67.5763 −30.9097 49.9986 4.6790 9.0069 59.9984 4.6790 9.0069 59.9984 3.9659 9.8626 59.9984 5.1178 8.6833 59.9984 3.5847 10.9103 59.9984 5.5950 8.4172 59.9984 3.5463 12.0265 59.9984 5.8487 8.3144 59.9984 3.6971 13.1346 59.9984 6.1024 8.2115 59.9984 3.9577 14.2207 59.9984 6.9225 8.0113 59.9984 4.3005 15.2821 59.9984 7.7617 7.9592 59.9984 4.7174 16.3215 59.9984 8.5955 8.0552 59.9984 5.1918 17.3336 59.9984 9.4128 8.2691 59.9984 5.7239 18.3155 59.9984 10.2137 8.5461 59.9984 6.3081 19.2589 59.9984 11.0008 8.8560 59.9984 6.9417 20.1668 59.9984 11.7825 9.1769 59.9984 7.6218 21.0389 59.9984 12.5614 9.5006 59.9984 8.3459 21.8782 59.9984 13.3403 9.8187 59.9984 9.1193 22.6818 59.9984 14.1274 10.1259 59.9984 9.9366 23.4442 59.9984 14.9228 10.4111 59.9984 10.7924 24.1546 59.9984 15.7264 10.6689 59.9984 11.6782 24.8183 59.9984 16.5382 10.8938 59.9984 12.6025 25.4299 59.9984 17.3610 11.0831 59.9984 13.5679 25.9894 59.9984 18.1920 11.2339 59.9984 14.5662 26.4941 59.9984 19.0286 11.3464 59.9984 15.5865 26.9329 59.9984 19.8678 11.4232 59.9984 16.6260 27.3059 59.9984 20.7125 11.4643 59.9984 17.6901 27.6158 59.9984 21.5545 11.4670 59.9984 18.7762 27.8544 59.9984 22.3993 11.4341 59.9984 19.8788 28.0245 59.9984 23.2385 11.3656 59.9984 20.9896 28.1232 59.9984 24.0778 11.2641 59.9984 22.0976 28.1506 59.9984 24.9088 11.1270 59.9984 23.2029 28.1040 59.9984 25.7371 10.9569 59.9984 24.3054 27.9861 59.9984 26.5571 10.7567 59.9984 25.4052 27.7968 59.9984 27.3690 10.5263 59.9984 26.4913 27.5445 59.9984 28.1726 10.2685 59.9984 27.5500 27.2318 59.9984 28.9679 9.9860 59.9984 28.5867 26.8616 59.9984 29.7551 9.6788 59.9984 29.5987 26.4392 59.9984 30.5313 9.3497 59.9984 30.5943 25.9675 59.9984 31.2992 9.0014 59.9984 31.5680 25.4464 59.9984 32.0562 8.6311 59.9984 32.5252 24.8814 59.9984 32.8077 8.2444 59.9984 33.4577 24.2807 59.9984 33.5482 7.8385 59.9984 34.3710 23.6444 59.9984 34.2805 7.4189 59.9984 35.2568 22.9780 59.9984 35.0018 6.9855 59.9984 36.1208 22.2868 59.9984 35.7176 6.5358 59.9984 36.9655 21.5683 59.9984 36.4225 6.0722 59.9984 37.7883 20.8250 59.9984 37.1219 5.5978 59.9984 38.5947 20.0625 59.9984 37.8103 5.1096 59.9984 39.3818 19.2781 59.9984 38.4904 4.6104 59.9984 40.1525 18.4773 59.9984 39.1624 4.1003 59.9984 40.9040 17.6627 59.9984 39.8261 3.5792 59.9984 41.6418 16.8344 59.9984 40.4844 3.0498 59.9984 42.3631 15.9979 59.9984 41.1316 2.5095 59.9984 43.0707 15.1532 59.9984 41.7734 1.9583 59.9984 43.7673 14.2975 59.9984 42.4042 1.4015 59.9984 44.4502 13.4308 59.9984 43.0295 0.8338 59.9984 45.1249 12.5586 59.9984 43.6494 0.2606 59.9984 45.7914 11.6782 59.9984 44.2610 −0.3209 59.9984 46.4441 10.7896 59.9984 44.8644 −0.9106 59.9984 47.0887 9.8928 59.9984 45.4623 −1.5057 59.9984 47.7250 8.9932 59.9984 46.0547 −2.1064 59.9984 48.3530 8.0854 59.9984 46.6389 −2.7152 59.9984 48.9756 7.1693 59.9984 47.2148 −3.3323 59.9984 49.5900 6.2505 59.9984 47.7880 −3.9522 59.9984 50.1961 5.3262 59.9984 48.3530 −4.5802 59.9984 50.7967 4.3965 59.9984 48.9098 −5.2138 59.9984 51.3919 3.4640 59.9984 49.4611 −5.8528 59.9984 51.9788 2.5260 59.9984 50.0069 −6.4946 59.9984 52.5575 1.5825 59.9984 50.5444 −7.1446 59.9984 53.1335 0.6363 59.9984 51.0765 −7.8001 59.9984 53.7012 −0.3127 59.9984 51.6031 −8.4611 59.9984 54.2635 −1.2644 59.9984 52.1242 −9.1248 59.9984 54.8230 −2.2188 59.9984 52.6398 −9.7940 59.9984 55.3742 −3.1787 59.9984 53.1472 −10.4687 59.9984 55.9200 −4.1387 59.9984 53.6491 −11.1462 59.9984 56.4603 −5.1041 59.9984 54.1483 −11.8263 59.9984 56.9951 −6.0722 59.9984 54.6392 −12.5120 59.9984 57.5245 −7.0431 59.9984 55.1274 −13.2004 59.9984 58.0456 −8.0168 59.9984 55.6101 −13.8916 59.9984 58.5612 −8.9932 59.9984 56.0873 −14.5882 59.9984 59.0713 −9.9750 59.9984 56.5591 −15.2876 59.9984 59.5705 −10.9624 59.9984 57.0281 −15.9897 59.9984 60.0669 −11.9498 59.9984 57.4916 −16.6946 59.9984 60.5551 −12.9453 59.9984 57.9523 −17.4022 59.9984 61.0378 −13.9409 59.9984 58.4049 −18.1125 59.9984 61.5123 −14.9393 59.9984 58.8547 −18.8283 59.9984 61.9813 −15.9431 59.9984 59.2990 −19.5469 59.9984 62.4448 −16.9496 59.9984 59.7351 −20.2682 59.9984 62.9001 −17.9589 59.9984 60.1657 −20.9950 59.9984 63.3526 −18.9710 59.9984 60.5853 −21.7273 59.9984 63.7969 −19.9857 59.9984 60.9939 −22.4651 59.9984 64.2385 −21.0033 59.9984 61.3944 −23.2084 59.9984 64.6801 −22.0208 59.9984 61.7811 −23.9571 59.9984 65.1216 −23.0356 59.9984 62.1596 −24.7113 59.9984 65.5687 −24.0476 59.9984 62.5271 −25.4711 59.9984 66.0240 −25.0597 59.9984 62.8919 −26.2308 59.9984 66.4820 −26.0662 59.9984 63.2566 −26.9932 59.9984 66.9290 −27.0810 59.9984 63.6351 −27.7502 59.9984 67.1279 −27.5994 59.9984 64.0410 −28.4935 59.9984 67.3267 −28.1177 59.9984 64.5183 −29.1956 59.9984 67.4502 −28.8116 59.9984 65.1381 −29.7715 59.9984 67.2198 −29.5000 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8.3678 29.1928 100.0000 11.1324 20.5535 100.0000 9.1934 29.7194 100.0000 11.9031 20.3286 100.0000 10.0518 30.1747 100.0000 12.6711 20.0817 100.0000 10.9322 30.5669 100.0000 13.4280 19.8130 100.0000 11.8455 30.8933 100.0000 14.1823 19.5250 100.0000 12.7863 31.1566 100.0000 14.9283 19.2205 100.0000 13.7435 31.3540 100.0000 15.6715 18.9079 100.0000 14.7089 31.4884 100.0000 16.4093 18.5815 100.0000 15.6770 31.5570 100.0000 17.1416 18.2496 100.0000 16.6479 31.5652 100.0000 17.8711 17.9095 100.0000 17.6216 31.5131 100.0000 18.5980 17.5585 100.0000 18.5925 31.4007 100.0000 19.3165 17.1992 100.0000 19.5551 31.2334 100.0000 20.0324 16.8317 100.0000 20.4986 31.0140 100.0000 20.7427 16.4532 100.0000 21.4284 30.7479 100.0000 21.4476 16.0637 100.0000 22.3444 30.4325 100.0000 22.1470 15.6661 100.0000 23.2495 30.0732 100.0000 22.8408 15.2601 100.0000 24.1409 29.6728 100.0000 23.5320 14.8433 100.0000 25.0130 29.2340 100.0000 24.2149 14.4181 100.0000 25.8660 28.7622 100.0000 24.8951 13.9848 100.0000 26.6970 28.2603 100.0000 25.5725 13.5460 100.0000 27.5088 27.7310 100.0000 26.2417 13.0989 100.0000 28.3070 27.1770 100.0000 26.9082 12.6464 100.0000 29.0886 26.5955 100.0000 27.5719 12.1884 100.0000 29.8538 25.9949 100.0000 28.2302 11.7249 100.0000 30.6053 25.3723 100.0000 28.8829 11.2559 100.0000 31.3431 24.7360 100.0000 29.5329 10.7814 100.0000 32.0644 24.0833 100.0000 30.1802 10.3014 100.0000 32.7720 23.4195 100.0000 30.8192 9.8132 100.0000 33.4632 22.7421 100.0000 31.4555 9.3195 100.0000 34.1406 22.0510 100.0000 32.0891 8.8231 100.0000 34.8098 21.3516 100.0000 32.7171 8.3185 100.0000 35.4653 20.6385 100.0000 33.3397 7.8056 100.0000 36.1098 19.9172 100.0000 33.9568 7.2900 100.0000 36.7434 19.1876 100.0000 34.5712 6.7689 100.0000 37.3714 18.4499 100.0000 35.1801 6.2423 100.0000 37.9913 17.7038 100.0000 35.7834 5.7102 100.0000 38.6001 16.9524 100.0000 36.3841 5.1727 100.0000 39.2035 16.1926 100.0000 36.9820 4.6296 100.0000 39.8014 15.4274 100.0000 37.5717 4.0838 100.0000 40.3911 14.6540 100.0000 38.1586 3.5325 100.0000 40.9753 13.8778 100.0000 38.7428 2.9758 100.0000 41.5512 13.0962 100.0000 39.3215 2.4163 100.0000 42.1245 12.3118 100.0000 39.8947 1.8513 100.0000 42.6894 11.5219 100.0000 40.4652 1.2836 100.0000 43.2490 10.7265 100.0000 41.0301 0.7103 100.0000 43.8030 9.9312 100.0000 41.5896 0.1316 100.0000 44.3515 9.1303 100.0000 42.1464 −0.4498 100.0000 44.8945 8.3240 100.0000 42.6977 −1.0367 100.0000 45.4348 7.5176 100.0000 43.2462 −1.6264 100.0000 45.9697 6.7058 100.0000 43.7865 −2.2216 100.0000 46.4990 5.8940 100.0000 44.3268 −2.8222 100.0000 47.0228 5.0767 100.0000 44.8589 −3.4256 100.0000 47.5440 4.2566 100.0000 45.3882 −4.0317 100.0000 48.0623 3.4338 100.0000 45.9121 −4.6433 100.0000 48.5752 2.6083 100.0000 46.4332 −5.2577 100.0000 49.0853 1.7827 100.0000 46.9488 −5.8775 100.0000 49.5900 0.9517 100.0000 47.4589 −6.5001 100.0000 50.0919 0.1207 100.0000 47.9663 −7.1254 100.0000 50.5883 −0.7131 100.0000 48.4682 −7.7562 100.0000 51.0820 −1.5496 100.0000 48.9674 −8.3870 100.0000 51.5729 −2.3889 100.0000 49.4583 −9.0233 100.0000 52.0584 −3.2281 100.0000 49.9493 −9.6624 100.0000 52.5411 −4.0728 100.0000 50.4347 −10.3069 100.0000 53.0210 −4.9176 100.0000 50.9147 −10.9514 100.0000 53.4955 −5.7651 100.0000 51.3919 −11.6014 100.0000 53.9673 −6.6125 100.0000 51.8636 −12.2514 100.0000 54.4362 −7.4655 100.0000 52.3354 −12.9069 100.0000 54.8998 −8.3185 100.0000 52.8016 −13.5624 100.0000 55.3605 −9.1742 100.0000 53.2651 −14.2234 100.0000 55.8158 −10.0326 100.0000 53.7259 −14.8844 100.0000 56.2683 −10.8911 100.0000 54.1812 −15.5454 100.0000 56.7181 −11.7523 100.0000 54.6365 −16.2118 100.0000 57.1624 −12.6162 100.0000 55.0835 −16.8811 100.0000 57.6013 −13.4829 100.0000 55.5306 −17.5503 100.0000 58.0374 −14.3496 100.0000 55.9694 −18.2250 100.0000 58.4707 −15.2190 100.0000 56.4027 −18.9024 100.0000 58.8985 −16.0912 100.0000 56.8306 −19.5853 100.0000 59.3237 −16.9633 100.0000 57.2502 −20.2737 100.0000 59.7488 −17.8382 100.0000 57.6616 −20.9676 100.0000 60.1739 −18.7131 100.0000 58.0620 −21.6642 100.0000 60.5990 −19.5881 100.0000 58.4515 −22.3691 100.0000 61.0269 −20.4602 100.0000 58.8300 −23.0795 100.0000 61.4602 −21.3269 100.0000 59.1975 −23.7980 100.0000 61.8990 −22.1936 100.0000 59.5650 −24.5139 100.0000 62.3324 −23.0603 100.0000 59.9545 −25.2270 100.0000 62.7465 −23.9406 100.0000 60.4125 −25.8989 100.0000 63.0948 −24.8512 100.0000 61.0049 −26.4420 100.0000 63.1058 −25.8029 100.0000 61.7537 −26.7052 100.0000 62.5161 −26.5544 100.0000 62.5161 −26.5544 100.0000

It will also be appreciated that the airfoil 52 disclosed in the above Table I may be scaled up or down geometrically for use in other similar turbine designs. Consequently, the coordinate values set forth in Table I may be scaled upwardly or downwardly such that the airfoil profile shape remains unchanged. A scaled version of the coordinates in Table I would be represented by X, Y and Z coordinate values of Table I, with the X, Y and Z non-dimensional coordinate values converted to inches, multiplied or divided by a constant number.

An important term in this disclosure is “profile”. The profile is the range of the variation between measured points on an airfoil surface and the ideal position listed in Table I.

The actual profile on a manufactured turbine rotor blade will be different than those in Table I and the design is robust to this variation meaning that mechanical and aerodynamic function are not impaired. As noted above, a + or −5% profile tolerance is used herein. The X, Y and Z values are all non-dimensionalized relative to the airfoil height.

The disclosed airfoil shape optimizes and is specific to the machine conditions and specifications. It provides a unique profile to achieve 1) interaction between other stages in the turbine 10; 2) aerodynamic efficiency; and 3) normalized aerodynamic and mechanical rotor blade or airfoil loadings. The disclosed loci of points defined in Table I allow the gas turbine 12 or any other suitable turbine to run in an efficient, safe and smooth manner. As also noted, any scale of the disclosed airfoil 52 may be adopted as long as 1) interaction between other stages in the pressure turbine 10; 2) aerodynamic efficiency; and 3) normalized aerodynamic and mechanical blade loadings are maintained in the scaled turbine. The airfoil 52 described herein thus improves overall gas turbine 12 efficiency. Specifically, the airfoil 52 provides a desired turbine efficiency lapse rate (ISO, hot, cold, part load, etc.). The airfoil 52 also meets all aeromechanics and stress requirements. The turbine rotor blade 50 described herein has very specific aerodynamic design requirements. Significant cross-functional design effort was require to meet these design goals. The airfoil 52 of the rotor blade 50, thus, is of a specific shape to meet aerodynamic, mechanical, and heat transfer requirements in an efficient and cost effective manner.

Further, it should be understood that exemplary embodiments of the present disclosure may include the entirety of the nominal airfoil profile set forth in Table I or portions thereof. Such portions may include a portion of the pressure side (or “pressure side portion”), a portion of the suction side (or “suction side portion”), and portions of both the pressure side and suction side.

Thus, exemplary embodiments of the present disclosure include a turbine rotor blade having an airfoil that includes a pressure side portion of a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of a pressure side as set forth in Table I. The Cartesian coordinate values of X, Y, and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil defined along the Z axis. The X and Y values of the pressure side are coordinate values that, when connected by smooth continuing arcs, define pressure side sections of the pressure side portion of the nominal airfoil profile at each Z coordinate value. The pressure side sections may be joined smoothly with one another to form the pressure side portion.

Exemplary embodiments of the present disclosure also include a turbine rotor blade having an airfoil that includes a suction side portion of a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of a suction side as set forth in Table I. The Cartesian coordinate values of X, Y, and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil defined along the Z axis. The X and Y values of the suction side are coordinate values that, when connected by smooth continuing arcs, define suction side sections of the suction side portion of the nominal airfoil profile at each Z coordinate value. The suction side sections may be joined smoothly with one another to form the suction side portion.

Exemplary embodiment of the present disclosure also include a turbine engine that has an airfoil, wherein the airfoil includes: a pressure side portion of a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of a pressure side as set forth in Table I; and a suction side portion of the nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of the suction side as set forth in Table I. The Cartesian coordinate values of X, Y, and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil defined along the Z axis. The X and Y values of the pressure side are coordinate values that, when connected by smooth continuing arcs, define pressure side sections of the pressure side portion of the nominal airfoil profile at each Z coordinate value. The pressure side sections may be joined smoothly with one another to form the pressure side portion. The X and Y values of the suction side are coordinate values that, when connected by smooth continuing arcs, define suction side sections of the suction side portion of the nominal airfoil profile at each Z coordinate value. The suction side sections may be joined smoothly with one another to form the suction side portion.

A height of each of the suction side portion and pressure side portion may be defined along the Z axis. The height of each of the suction side portion and pressure side portion may be less than or equal to the height of the airfoil. According to exemplary embodiments, the height of the suction side portion and the height of the pressure side portion may be substantially the same. According to exemplary embodiments, the suction side portion and the pressure side portion may each start at a common distance relative to a base of the airfoil and extend toward a tip of the airfoil. According to exemplary embodiments, the height of the suction side portion and/or the height of the pressure side portion may each be equal to or greater than 50% of the height of the airfoil. According to other embodiments, the height of the suction side portion and/or the height of the pressure side portion may each be equal to or greater than 75% of the height of the airfoil. According to other embodiments, the height of the suction side portion and the height of the pressure side portion may each be equal to 100% of the height of the airfoil.

As already described, the turbine engine may include a combustion or gas turbine that has a compressor, combustor, and turbine. The airfoil of the present disclosure may be configured as the airfoil of a turbine rotor blade, i.e., a rotor blade within the turbine of the gas turbine. More specifically, the airfoil of the present disclosure may be the airfoil of a turbine rotor blade that is configured to function as a first stage rotor blade in the turbine. In such cases, the turbine includes a first stage that has a row of circumferentially spaced nozzles and a row of circumferentially spaced rotor blades, with the airfoil of the present disclosure being configured as the airfoil of each of the rotor blades within the first stage row of rotor blades.

This written description uses examples to disclose the invention, including the best mode, and also to enable any person skilled in the art to practice the invention, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the invention is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they have structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal languages of the claims. 

What is claimed:
 1. A turbine rotor blade including an airfoil that comprises a pressure side portion of a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of a pressure side as set forth in Table I, wherein: the Cartesian coordinate values of X, Y, and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil defined along a Z axis; the X and Y values of the pressure side are coordinate values that, when connected by smooth continuing arcs, define pressure side sections of the pressure side portion of the nominal airfoil profile at each Z coordinate value; and the pressure side sections are joined smoothly with one another to form the pressure side portion.
 2. The turbine rotor blade of claim 1, wherein the nominal airfoil profile lies in an envelope within +/− 5% of a chord length in a direction normal to any of the pressure side sections.
 3. The turbine rotor blade of claim 1, wherein a height of the pressure side portion is defined along the Z axis, and wherein the height of the pressure side portion is less than or equal to the height of the airfoil.
 4. The turbine rotor blade of claim 3, wherein the height of the pressure side portion is equal to or greater than 50% of the height of the airfoil; and wherein the height of the airfoil is between 3 and 10 inches.
 5. The turbine rotor blade of claim 3, wherein the height of the pressure side portion is equal to or greater than 75% of the height of the airfoil; and wherein the turbine rotor blade is configured to function as a first stage turbine rotor blade in a turbine.
 6. The turbine rotor blade of claim 3, wherein the height of the pressure side portion is equal to 100% of the height of the airfoil.
 7. A turbine rotor blade including an airfoil that comprises a suction side portion of a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of a suction side as set forth in Table I, wherein: the Cartesian coordinate values of X, Y, and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil defined along a Z axis; the X and Y values of the suction side are coordinate values that, when connected by smooth continuing arcs, define suction side sections of the suction side portion of the nominal airfoil profile at each Z coordinate value; and the suction side sections are joined smoothly with one another to form the suction side portion.
 8. The turbine rotor blade of claim 7, wherein the nominal airfoil profile lies in an envelope within +/− 5% of a chord length in a direction normal to any of the suction side sections.
 9. The turbine rotor blade of claim 7, wherein a height of the suction side portion is defined along the Z axis, and wherein the height of the suction side portion is less than or equal to the height of the airfoil.
 10. The turbine rotor blade of claim 9, wherein the height of the suction side portion is equal to or greater than 50% of the height of the airfoil; and wherein the height of the airfoil is between 3 and 10 inches.
 11. The turbine rotor blade of claim 9, wherein the height of the suction side portion is equal to or greater than 75% of the height of the airfoil; and wherein the turbine rotor blade is configured to function as a first stage turbine rotor blade in a turbine.
 12. The turbine rotor blade of claim 9, wherein the height of the suction side portion is equal to 100% of the height of the airfoil.
 13. A turbine engine that includes an airfoil that comprises: a pressure side portion of a nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of a pressure side as set forth in Table I; a suction side portion of the nominal airfoil profile substantially in accordance with Cartesian coordinate values of X, Y, and Z of the suction side as set forth in Table I wherein: the Cartesian coordinate values of X, Y, and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil defined along a Z axis; the X and Y values of the pressure side are coordinate values that, when connected by smooth continuing arcs, define pressure side sections of the pressure side portion of the nominal airfoil profile at each Z coordinate value, the pressure side sections being joined smoothly with one another to form the pressure side portion; and the X and Y values of the suction side are coordinate values that, when connected by smooth continuing arcs, define suction side sections of the suction side portion of the nominal airfoil profile at each Z coordinate value, the suction side sections being joined smoothly with one another to form the suction side portion.
 14. The turbine engine of claim 13, wherein the turbine engine comprises a gas turbine having a compressor, a combustor, and a turbine; and wherein the airfoil comprises a rotor blade in the turbine.
 15. The turbine engine of claim 14, wherein the rotor blade is a first stage rotor blade in the turbine.
 16. The turbine engine of claim 14, wherein the nominal airfoil profile lies in an envelope within +/−20% of a thickness of a trailing edge of the airfoil in a direction normal to any of the suction side and pressure side sections; and wherein: a height of the suction side portion is defined along the Z axis, and wherein the height of the suction side portion is less than or equal to the height of the airfoil; and a height of the pressure side portion is defined along the Z axis, and wherein the height of the pressure side portion is less than or equal to the height of the airfoil.
 17. The turbine engine of claim 16, wherein the height of the suction side portion and the height of the pressure side portion are substantially the same; wherein the height of the suction side portion and the height of the pressure side portion are each equal to or greater than 50% of the height of the airfoil; and wherein the suction side portion and the pressure side portion start at a common distance relative to a base of the airfoil and extend toward a tip of the airfoil.
 18. The turbine engine of claim 17, wherein the height of the suction side portion and the height of the pressure side portion are each equal to or greater than 75% of the height of the airfoil.
 19. The turbine engine of claim 17, wherein the height of the suction side portion and the height of the pressure side portion are each equal to or greater than 100% of the height of the airfoil.
 20. The turbine engine of claim 17, wherein the turbine comprises a first stage that includes a row of first stage nozzles adjacent to a row of first stage rotor blades; and wherein a plurality of first stage rotor blades contained within the row of first stage rotor blades comprises the airfoil. 